30 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA
(c) Deduce from (a) and (b) that
A ^ I/log 4, ^ 5 I/log 2
N
(use the fact that £ 1/(N + r) tends to log 2 as N -»• + <x>).
r=l
(d) Take an = Iog2(2n - 1) - [Iog2(2w - 1)]. Show that we have A = I/log 4 and
p, = i/iog 2 for this sequence. (Observe that, if N = 2P, the set {aif..., <ZN} is the same
as the set {CN ,..., c2N -1} where cn = Iog2 n — [Iog2 «].) (Cf. Section 13.21, Problem 15.)
7. Let E be a metrizable space. Show that a nonempty closed subset S of E is the support
of a lower semicontinuous (resp. continuous) real-valued function ^0 on E if and only
if S is the closure of its interior.
8. Let E be a metrizable space and / a mapping of E into ft. The lower semicontinuous
regularization of/is defined to be the upper envelope of the continuous functions g on
E such that g<*f.
(a) For each x e E we denote by lim inf f(y) the greatest lower bound of the numbers
y-*x
lim inf/(>>„), for all sequences (yn) tending to x Show that the function ;eh->-lim inf/0>)
n-»oo y-*x
is the lower semicontinuous regularization of/.
(b) Suppose that E is an open set in R. For each x e E, let lim inf f(y) denote the
y-»x, y£x
greatest upper bound of the numbers lim inf/OB) for all sequences (yn) such that
n-*oo
yn -»• x and yn ^> x for all n. Show that the set of all x e E such that
lim inf f(y) > lim inff(y)
y-*x, y&x y-+x
is at most denumerable. (For each pair (p, q) of rational numbers with p>g, show
that the set of points x e E such that
lim inf f(y) >p>q>\im Mf(y)
y-+xt y&x y-*x
is at most denumerable, by following the method of Section 3.9, Problem 3.)
9. A Dirichlet series is a series whose general term is of the form ane~*n5, where (An)
is an increasing sequence of real numbers, tending to + oo, and (an) is any sequence
of complex numbers, and s is complex.
(a) Show that if the series is convergent for 5- = s0, then it is uniformly convergent in
the angular sector consisting of the points s = s0 + pew with p ^> 0 and — a £ 6 <i a,
where a is such that 0 < a < 7r/2. (Reduce to the case s0 — 0; show first that if
9ts = or, then
\e-**^e~**\ g |jio-H*""-«-**)
Jb
e~~xs dx. For each
a
integer m and each n^m, put S™, „ — am + am+i H-------h an and use the identity
(AbeFs partial summation formula)reater than or equal